Forbidden paths and cycles in ordered graphs and matrices

János Pach, G. Tardos

Research output: Contribution to journalArticle

28 Citations (Scopus)

Abstract

At most how many edges can an ordered graph of n vertices have if it does not contain a fixed forbidden ordered subgraph H? It is not hard to give an asymptotically tight answer to this question, unless H is a bipartite graph in which every vertex belonging to the first part precedes all vertices belonging to the second. In this case, the question can be reformulated as an extremal problem for zero-one matrices avoiding a certain pattern (submatrix) P. We disprove a general conjecture of Füredi and Hajnal related to the latter problem, and replace it by some weaker alternatives. We verify our conjectures in a few special cases when P is the adjacency matrix of an acyclic graph and discuss the same question when the forbidden patterns are adjacency matrices of cycles. Our results lead to a new proof of the fact that the number of times that the unit distance can occur among n points in the plane is O(n 4/3).

Original languageEnglish
Pages (from-to)359-380
Number of pages22
JournalIsrael Journal of Mathematics
Volume155
DOIs
Publication statusPublished - 2006

Fingerprint

Adjacency Matrix
Cycle
Path
Disprove
Extremal Problems
Graph in graph theory
Bipartite Graph
Subgraph
Verify
Unit
Alternatives
Zero
Vertex of a graph

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Forbidden paths and cycles in ordered graphs and matrices. / Pach, János; Tardos, G.

In: Israel Journal of Mathematics, Vol. 155, 2006, p. 359-380.

Research output: Contribution to journalArticle

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